# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Numerical calculation of incomplete gamma functions by the trapezoidal rule. (English) Zbl 0593.65017

The trapezoidal rule is applied to the numerical calculation of the known integral representation of the complementary incomplete gamma function

${\Gamma }\left(a,x\right)=\left(2{e}^{-x}{x}^{a}/{\Gamma }\left(1-a\right)\right){\int }_{0}^{+\infty }{e}^{-{u}^{2}}{u}^{-2a+1}/x+\phantom{\rule{1.em}{0ex}}{u}^{2}\phantom{\rule{1.em}{0ex}}du$

in the region $a<-1$ and $x>0$. This application of the trapezoidal rule is not standard since the function ${u}^{-2a+1}{\left(x+{u}^{2}\right)}^{-1}$ in the integrand is not even, but numerical tests show that using the rule may still be convenient. The explanation of this surprising fact requires a careful investigation of the related Euler-Maclaurin formula and expecially of its remainder term. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with good accuracy in the stated region. The explained method can be applied to the numerical evaluation of other important special functions.

##### MSC:
 65D20 Computation of special functions, construction of tables 65B15 Euler-Maclaurin formula (numerical analysis) 33B15 Gamma, beta and polygamma functions
##### References:
 [1] Allasia, G., Besenghi, R.: Sul calcolo numerico delle funzioni gamma e digamma mediante la formula del trapezio. Boll. Unione Mat. Ital. (to appear) [2] Davis, P.J., Rabinowitz, P.: Methods of numerical integration (2nd Ed.). New York: Academic Press 1984 [3] Erd?ly, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. New York: McGraw-Hill 1953 [4] Gatteschi, L.: Funzioni speciali. Torino: U.T.E.T. 1973 [5] Gautschi, W.: Un procedimento di calcolo per le funzioni gamma incomplete. Rend. Semin. Mat. Torino37, 1-9 (1979) [6] Gautschi, W.: A computational procedure for incomplete gamma functions. ACM Trans. Math. Software5, 466-481 (1979) · Zbl 0429.65011 · doi:10.1145/355853.355863 [7] Gautschi, W.: Algorithm 542-Incomplete gamma functions. ACM Trans. Math. Software5, 482-489 (1979) · Zbl 0434.65007 · doi:10.1145/355853.355864 [8] Hardy, G.H.: Divergent series. Oxford: University Press 1949 [9] Hunter, D.B.: The calculation of certain Bessel functions. Math. Comput.18, 123-128 (1964) · doi:10.1090/S0025-5718-1964-0158104-3 [10] Hunter, D.B.: The evaluation of a class of functions defined by an integral. Math. Comput.22, 440-444 (1968) · doi:10.1090/S0025-5718-68-99874-8 [11] Lindel?f, E.: Le calcul des r?sidus. New York: Chelsea 1947 [12] Luke, Y.L.: The special functions and their approximations. New York: Academic Press 1969 [13] Luke, Y.L.: Mathematical functions and their approximations. New York: Academic Press 1975 [14] McNamee, J.: Error bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae. Math. Comput.18, 368-381 (1964) · doi:10.1090/S0025-5718-1964-0185804-1 [15] Pagurova, V.I.: Tables of the Exponential integral ${E}_{r}\left(x\right)=\underset{1}{\overset{x}{\int }}{e}^{-xu}{u}^{-r}du$ . New York: Pergamon Press 1961 [16] Tricomi, F.G.: Sulla funzione gamma incompleta. Ann. Mat. Pura Appl.31, 263-279 (1950) · Zbl 0040.18401 · doi:10.1007/BF02428264